Take a cab into Rome from Fiumicino airport and the odds are that the driver will hurtle you up the Via Ostiense, across the Aventine hill, and bring you out in front of the most breathtaking welcome to any European capital.

Across the Circus Maximus, the remains of the palace of the ruthless Roman emperor Septimius Severus rear up like a Grand Canyon. At the eastern end, there is a gate in a fence and beyond lies a path winding through an eery, secret corner of the city.

"It was closed off in the 1960s," said Giovanna Tedone, shutting the gate behind her. Halfway along the path, which runs beneath the palace ruins, stands a warden's hut smothered with graffiti. Nearby, a broken floodlight lies by a rusting gate.

The reason no one below the age of about 60 has seen at close hand Rome's most massively imposing classical remains is that it is too risky to let them in.

"Look up there," said Ms Tedone, an architect with the state's archaeological service in Rome. She was pointing to a huge crack in the south wall of the stadium built by the emperor Domitian. "All you need is for a plant to get in there, or some ice and, after a while, the wall could fall at any moment."

Everyone has heard of sinking Venice, but only a restricted circle of academics wring their hands over crumbling Rome. Yet, for lack of funds, the city's ancient centre is inexorably deteriorating.

Now, though, the issue has been given new urgency by climate change. One night last November, a wall on the Palatine Hill collapsed. It had been built in the 16th or 17th centuries in the landscaped gardens overlooking the Forum created by the aristocratic Farnese family, and gave way after days of torrential rain of a kind that has become increasingly common in Rome.

Angelo Bottini, Rome's chief government archaeologist, said one of his first responses was to call meteorologists: "They confirmed that the Roman climate was getting rainier, and that last winter was one of the rainiest on record."

The wall's collapse raised an alarming question: if the rain could bring down an apparently solid, 400-year-old structure, what might it do to 2000-year-old buildings suspected to be unstable?

"If you think that an ordinary house, with a roof, needs to be kept up, you can imagine what is needed to maintain a monument that has no roof and maybe only partial walls," said Prof Bottini.

His next move was to hire a renowned structural engineer, Giorgio Croci, to survey the Palatine, to establish which monuments were most at risk.

"We don't know if those that appear to be solid really are," Prof Bottini said. The Palatine is honeycombed with cavities. Some are natural. Some are man-made: the results of tunnelling and digging over many centuries.

What is more, instead of demolishing buildings that they no longer wanted, the ancient Romans had the vexing habit of using them as the basis for others. So many stuctures that look solid could rest on shaky foundations.

As the authorities wait for the engineer's report, emergency repairs are being carried out on two areas where the risks are all too visible. One is Tiberius's palace, which cascades down the north side of the Palatine, overlooking the Forum. The other is the Domus Aurea, Nero's palace, beyond the Colosseum.

Part of this vast complex was opened to the public in 2000. In December, after more heavy rain, it was abruptly closed.

Trajan, the first non-Italian emperor, who came to power 30 years after Nero's death, filled in the Domus Aurea with earth to build baths on top. These too were later filled in, so that both structures today lie underneath a park.

"When it rains, the water from above goes straight down," said Prof Bottini. "We have a simple project. It involves removing all the earth, waterproofing the monuments underneath and then putting back the earth."

A small, raised walkway is to be built so that a tiny part of the Domus Aurea can be reopened. The €4m (£2.7m) scheme is due to be finished in three years. A further €8.5m has been set aside for the Palatine with €3.5m for smaller emergency projects elsewhere.

But, depending on Prof Croci's findings, much more cash may be needed, and that could switch attention to perhaps the oddest aspect of Rome's disintegration.

Substantial funds are potentially available for its conservation. But the government has decided not to tap into them. Entrance to the Forum and the Palatine Hill is free. It is arguably the world's greatest tourist bargain.

"We've thought about it so many times. It's lovely. It's right," sighed Ms Tedone, before adding a pregnant "But ..."

No one knows how many visitors there are. Ms Tedone estimates 10-15,000 a day. If that is right and they each paid only a euro, it would raise more than €45m a year. That is three times the budget available for preserving the most important site handed down from the classical world.